Polynomial values and coefficient parameters
To evaluate a polynomial, substitute the input into every term and keep signed inputs in brackets. To find a coefficient parameter, turn each stated polynomial value or remainder into an exact equation
\[P(a)=c\quad\text{means substitute }x=a\text{ and set the result equal to }c\]
- Substitute in every term Replace every occurrence of the variable by the stated input \(\displaystyle P(x)=2x^2-3x+1\Rightarrow P(2)=2(2)^2-3(2)+1\)
- Keep brackets around signed inputs Use brackets so the sign is included when a power is evaluated \(\displaystyle (-3)^2=9,\qquad -3^2=-9\)
- Evaluate powers first Find each required power before multiplying by its coefficient and combining terms \(\displaystyle P(-1)=2(-1)^2-3(-1)+1=6\)
- Turn conditions into equations For an unknown coefficient, substitute the stated input and solve the resulting exact equation \(\displaystyle P(x)=x^2+px+1,\quad P(2)=11\Rightarrow4+2p+1=11\Rightarrow p=3\)
Worked example
Evaluate the polynomial at the stated input
\[P(t)=3t^3-2t^2+4t-5,\qquad P(-1)\]
- \(\displaystyle P(-1)=3(-1)^3-2(-1)^2+4(-1)-5\)Replace every t by -1 and keep brackets around the signed input
- \(\displaystyle (-1)^3=-1,\qquad (-1)^2=1\)Evaluate the powers before combining the terms
- \(\displaystyle P(-1)=3(-1)-2(1)+4(-1)-5=-3-2-4-5=-14\)Multiply each power by its coefficient and add the resulting terms
Common mistakes
Leaving out a term
✕\(\displaystyle P(-2)=2(-2)^3+4=-12\)✓\(\displaystyle P(-2)=2(-2)^3-3(-2)+4=-6\)
The input must be substituted into every non-zero term of the polynomialLosing the sign in a power
✕\(\displaystyle (-3)^2=-9\)✓\(\displaystyle (-3)^2=9\)
The brackets make the negative input part of the square, and two negative factors give a positive resultTreating a parameter as the input
✕\(\displaystyle P(x)=x^2+px+1,\quad P(2)=2^2+2^2+1=9\)✓\(\displaystyle P(x)=x^2+px+1,\quad P(2)=2^2+2p+1=5+2p\)
The input replaces x, while p remains the unknown coefficient parameterDifficulty levels
introEvaluate an integer-coefficient polynomial at an integer input
standardEvaluate at an exact rational input or find one coefficient from a stated value
challengeEvaluate an exact surd or recover two coefficients from two exact conditions
examHandle exact value, zero or remainder information and verify the resulting polynomial
Questions about this topic
What does P(a) mean?+
P(a) is the value of the polynomial when every occurrence of the variable is replaced by a.
Why are brackets needed around a negative input?+
Brackets show that the negative value is the base of the power. Squaring negative two gives four; squaring two and then negating gives negative four
How is a coefficient parameter found from P(a)?+
Substitute a into every term, set the expression equal to the stated value, and solve the resulting equation for the parameter
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 6: Applications of differentiation
In this chapter
Differentiate polynomialsChallenge enrichment: rational powers and derivative domainsDifferentiate natural exponential functions and solve simple derivative conditionsDifferentiate sine, cosine and tangent functions and solve simple derivative conditionsDifferentiate a composite function by the chain ruleDifferentiate a product or quotient of two functionsEvaluate and determine function rules6B Differentiation rules: Find the maximal domain of an algebraic functionExpand and factor polynomial expressionsSimplify and combine algebraic fractions6B Differentiation rules: Find the equation of a line from its graph or from its intercept and slopeAnalyse polynomial curves, stationary points and extremaUse a derivative to analyse and recognise the graph of a function6C Tangents, normals and curve sketching: Evaluate a polynomial or recover a coefficient parameter6C Tangents, normals and curve sketching: Move between roots, factors, polynomial rules and equationsRecognise and classify a polynomialConnect a factorised higher-degree rule and its graphAnalyse a cubic graph or determine its ruleWork with sets and interval notation6C Tangents, normals and curve sketching: Find the maximal domain of an algebraic function6C Tangents, normals and curve sketching: Find the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular lineSolve a maximum or minimum problem by differentiationSet up and solve a practical linear equationRearrange literal equations and recognise equivalent line forms6D Optimisation: Find the maximal domain of an algebraic function6D Optimisation: Evaluate a polynomial or recover a coefficient parameter6D Optimisation: Move between roots, factors, polynomial rules and equationsCalculate and interpret average and instantaneous rates from a modelVolume of cones, pyramids and spheresVolume of prisms and cylindersPerimeter and area of standard shapesSimilar figures and scale factorsFind a side of a right-angled triangleAnalyse straight-line motion through displacement, velocity and acceleration6F Kinematics: Evaluate a polynomial or recover a coefficient parameter